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1 1 IX Mathematics Chapter 8: Quadrilaterals Chapter Notes Top Definitions 1. A quadrilateral is a closed figure obtained by joining four points (with no three points collinear) in an order. 2. A diagonal is a line segment obtained on joining the opposite vertices. 3. Two sides of a quadrilateral having no common end point are called its opposite sides. 4. Two angles of a quadrilateral having common arm are called its adjacent angles. 5. Two angles of a quadrilateral not having a common arm are called its opposite angles. 6. A trapezium is quadrilateral in which one pair of opposite sides are parallel. 7. In the non parallel sides of trapezium are equal, it is known as isosceles trapezium. 8. A parallelogram is a quadrilateral in which both the pairs of opposite sides are parallel. 9. A rectangle is a quadrilateral whose each angle is A rhombus is quadrilateral whose all the sides are equal. 11. A square is a quadrilateral whose all sides are equal and each angle is A kite is a quadrilateral in which two pairs of adjacent sides are equal. Top Concepts 1. Properties of parallelogram: i The opposite sides of a parallelogram are parallel. ii A diagonal of a parallelogram divides it in two congruent triangles. iii The opposite sides of a parallelogram are equal. iv The opposite angles of a parallelogram are equal. v The consecutive angles (conjoined angles) of a parallelogram are supplementary.

2 2 vi The diagonals of a parallelogram bisect each other. 2. A diagonal of a parallelogram divides the parallelogram into two congruent triangles. 3. In a parallelogram opposite sides are equal. 4. If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram. 5. In a parallelogram opposite angles are equal. 6. If in quadrilateral, each pair of opposite angles is equal, then it is a parallelogram. 7. The diagonals of a parallelogram bisect each other. 8. If the diagonals of a quadrilateral bisect other, then it is a parallelogram. 9. A quadrilateral is a parallelogram, if a pair of opposite sides is equal and parallel. 10. Square, rectangle and rhombus are all parallelograms. 11. Kite and trapezium are not parallelogram. 12. A square is a rectangle. 13. A square is a rhombus. 14. A parallelogram is a trapezium. 15. Every rectangle is a parallelogram; therefore, it has all the properties of a parallelogram. Additional properties of a rectangle are: i All the (interior) angles of are rectangle are right angles. ii The diagonals of a rectangle are equal. 16. Every rhombus is a parallelogram; therefore, it has all the properties of a parallelogram. Additional properties of a rhombus are: i All the sides of rhombus are equal. ii The diagonals of a rhombus intersect at right angles. iii The diagonals bisect the angles of a rhombus. 17. Every square is a parallelogram; therefore, it has all the properties of a parallelogram. Additional properties of a rhombus are: i All the sides are equal ii All the angles are equal to 90 each iii Diagonals are equal

3 3 iv Diagonal bisect each other at right angle v Diagonals bisects the angles of vertex 18. Sum of all the angles of a quadrilateral is Mid Point Theorem (Basic Proportionality Theorem): The line segment joining the mid point of any two sides of a triangle is parallel to the third sides and equal to half of it. 20. Converse of mid-point theorem: The line drawn through the mid-point of one side of a triangle parallel to the another side, bisects the third side. 21. If there are three or more parallel lines and the interests made by them on a transversal are equal, then the corresponding intercepts on any other transversal are also equal. 22. A quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order is a parallelogram. Top Diagrams 1. A quadrilateral ABCD. 2. A trapezium ABCD with sides AB DC and non parallel sides AD and BC. 3. A parallelogram ABCD in which AB DC and AD BC.

4 4 a 4. A rectangle ABCD with AD BC, AB DC and A = 90 = B = C = D. 5. A rhombus ABCD with AB = BD = CD = DA. 6. A square ABCD in which AB = BC = CD, = DA and A = B = C = D = A kite ABCD with AB = AD and BC = CD

5 5 8. Diagonal properties of special parallelograms: Properties Parallelogram Rectangle Rhombus Square Diagonals bisect each other Diagonals are equal Diagonals bisect vertex angles Diagonals are perpendicular to each other Diagonals from 4 equal triangle Diagonals from 4 congruent triangle 9. The relations between special parallelograms can be represented by a Veen- diagram.

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